Sentence examples for be a normalized duality from inspiring English sources

Let be a real Banach space with the dual space and be a normalized duality mapping defined by (1.1).

Let E be a real Banach space and J : E → 2 E ∗ be a normalized duality mapping.

Let E be a real Banach space, and let J : E → 2 E ∗ be a normalized duality mapping.

Let E be a uniformly smooth real Banach space, and let J : E → 2 E ∗ be a normalized duality mapping.

Let D be a nonempty subset of C. Let Π be a retraction of C onto D and let J be a normalized duality mapping on X.

Definition 2.4 Let A, B : X → X, H : X × X → X, η : X × X → X be three single-valued mappings and J : X → 2 X ∗ be a normalized duality mapping.

In addition, we assume the following property ( P ) : For ϵ > 0, there exists a λ > 0 such that the inclusion 0 ∈ T x + λ C x + ε J x. has no solution in D ( T ) ∩ D ( C ) ∩ Ω, where Ω is a bounded open set in X and J is a normalized duality operator.

(1) If E is a uniformly smooth real Banach space, then J is uniformly continuous on each bounded subset of E.   (2) If E is a uniformly smooth real Banach space, then J ∗ : E ∗ → 2 E is a normalized duality mapping on E ∗, then J − 1 = J ∗, ( J ∗ ) J = I E and J J J ∗ ) = I E ∗, where on I E and I E ∗ are the identity mappings on E and E ∗, respectively.

If E is a uniformly smooth real Banach space, then J is uniformly continuous on each bounded subset of E. If E is a uniformly smooth real Banach space, then J ∗ : E ∗ → 2 E is a normalized duality mapping on E ∗, then J − 1 = J ∗, ( J ∗ ) J = I E and J J J ∗ ) = I E ∗, where on I E and I E ∗ are the identity mappings on E and E ∗, respectively.

In particular, (J=J_{2}) is called a normalized duality mapping and (J_{q}(x)=Vert xVert ^{q-2}J_{2}(x)) for (xneq0).

A map (J Xrightarrow2^{ X^) defined by (Ju:= {u^in X^: langle u,u^rangle=Vert uVert Vert u^Vert, Vert uVert =Vert u^Vert }) is called a normalized duality map on X, where (langlecdot,cdot rangle) denotes the duality pairing between elements of X and (X^).